The Geometry of Relevant Implication Alasdair Urquhart University of Toronto urquhart@cs.toronto.edu Abstract This paper is a continuation of earlier work by the author on the connection between the logic KR and projective geometry. It contains a simplified con- struction of KR model structures; as a consequence, it extends the previous results to a much more extensive class of projective spaces and the correspond- ing modular lattices. 1 The Logic KR The logic KR occupies a rather unusual place in the family of relevant logics. In fact, it is questionable whether it should even be classified as a relevant logic, since it is the result of adding to R the axiom ex falso quodlibet , that is to say, ( A ∧¬ A ) → B . This is of course one of the paradoxes of material implication that relevant logics were devised specifically to avoid, a paradox of consistency. The other type of paradox is a paradox of relevance, of which the paradigm case is the weakening axiom A → ( B → A ). The surprising thing about KR is that although it contains the first type of paradox, it avoids the second, contrary to what we might at first suspect. In fact, it is a complex and highly non-trivial system. The credit for its initial investigation belongs to Adrian Abraham, Robert K. Meyer and Richard Routley [12]. The model theory for KR is elegantly simple. The usual ternary relational semantics for R includes an operation ∗ designed to deal with the truth condition for negation = ¬ A ⇔ x ∗ �| x | = A. The effect of adding ex falso quodlibet to R is to identify x and x ∗ ; this in turn has a notable effect on the ternary accessibility relation. The postulates for an R model structure include the following implication: Rxyz ⇒ ( Ryxz & Rxz ∗ y ∗ ) . Vol. \ jvolume No. \ jnumber \ jyear IFCoLog Journal of Logics and their Applications
Alasdair Urquhart The result of the identification of x and x ∗ is that the ternary relation in a KR model structure (KRms) is totally symmetric . In detail, a KRms K = � S, R, 0 � is a 3-place relation R on a set containing a distinguished element 0, and satisfying the postulates: 1. R 0 ab ⇔ a = b ; 2. Raaa ; 3. Rabc ⇒ ( Rbac & Racb ) (total symmetry); 4. ( Rabc & Rcde ) ⇒ ∃ f ( Rad f & Rfbe ) (Pasch’s postulate). The result of adding the weakening axiom A → ( B → A ) to R is a collapse into classical logic. The addition of ( A ∧ ¬ A ) → B does not result in such a collapse – but is the result a trivial or uninteresting system? This is very far from the case, as we shall see in the next section. 2 KR and modular lattices Given a KR model structure K = � S, R, 0 � , we can define an algebra A ( K ) as follows: Definition 2.1. The algebra A ( K ) = �P ( S ) , ∩ , ∪ , ¬ , ⊤ , ⊥ , t, ◦� is defined on the Boolean algebra �P ( S ) , ∩ , ∪ , ¬ , ⊤ , ⊥� of all subsets of S , where ⊤ = S, ⊥ = ∅ , t = { 0 } , and the operator A ◦ B is defined by A ◦ B = { c | ∃ a ∈ A, b ∈ B ( Rabc ) } . The algebra A ( K ) is a De Morgan monoid [1],[3] in which A ∩ ¬ A = ⊥ ; we shall call any such algebra a KR -algebra . Hence the fusion operator A ◦ B is associative, commutative, and monotone. In addition, it satisfies the square-increasing property, and t is the monoid identity: A ◦ ( B ◦ C ) = ( A ◦ B ) ◦ C, A ◦ B = B ◦ A, ( A ⊆ B ∧ C ⊆ D ) ⇒ A ◦ C ⊆ B ◦ D, A ⊆ A ◦ A, A ◦ t = A. 2
Geometry of Relevant Implication In what follows, we shall assume basic results from the theory of De Morgan monoids, referring the reader to the expositions in Anderson and Belnap [1] and Dunn and Restall [3] for more background. In a KR -algebra, we can single out a subset of the elements that form a lattice; this lattice plays a key role in the analysis of the logic KR . Definition 2.2. Let A be a KR -algebra. The family L ( A ) is defined to be the elements of A that are ≥ t and idempotent, that is to say, a ∈ L ( A ) if and only if a ◦ a = a and t ≤ a . If K is a KR model structure, then we define L ( K ) to be L ( A ( K )) . The following lemma provides a useful characterization of the elements of L ( A ); it is based on some old observations of Bob Meyer. Lemma 2.3. Let A be a KR -algebra. Then the following conditions are equivalent: 1. a ∈ L ( A ) ; 2. a = ( a → a ) ; 3. ∃ b [ a = ( b → b )] . Proof. ( 1 ⇒ 2 ⇒ 3 ): Since t ≤ a , we have t ≤ ( a → a ) → a , t ◦ ( a → a ) ≤ a , hence ( a → a ) ≤ a . Since a ◦ a ≤ a , a ≤ ( a → a ), so a = ( a → a ), proving the second and hence the third condition. ( 3 ⇒ 1 ): First, we have t ≤ ( b → b ) = a . Second, ( b → b ) ≤ ( b → b ) → ( b → b ), so ( b → b ) ◦ ( b → b ) ≤ ( b → b ), that is to say, a ◦ a ≤ a , so a ◦ a = a . If K = � S, R, 0 � is a KR model structure, then a subset A of S is a linear subspace if it satisfies the condition ( a, b ∈ A ∧ Rabc ) ⇒ c ∈ A. A lattice is modular if it satisfies the implication x ≥ z ⇒ x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ z. For background on modular lattice theory, the reader can consult the texts of Birkhoff [2] or Grätzer [6]. We require a few basic lattice-theoretic definitions here. A chain in a lattice L is a totally ordered subset of L ; the length of a finite chain C is | C | − 1. A chain C in a lattice L is maximal if for any chain D in L , if C ⊆ D then C = D . If L is 3
Alasdair Urquhart a lattice, a, b ∈ L and a ≤ b , then the interval [ a, b ] is defined to be the sublattice { c : a ≤ c ≤ b } . Let L be a lattice with least element 0. We define the height function: for a ∈ L , let h ( a ) denote the length of a longest maximal chain in [0 , a ] if there is a finite longest maximal chain; otherwise put h ( a ) = ∞ . If L has a largest element 1, and h (1) < ∞ , then L has finite height . Let L be a modular lattice with 0 of finite height. Then for a ∈ L , h ( a ) is the length of any maximal chain in [0 , a ]. In addition, the height function in L satisfies the condition h ( a ) + h ( b ) = h ( a ∧ b ) + h ( a ∨ b ) , for all a, b ∈ L . For a lattice of finite height, this last condition is equivalent to modularity. These results are proved in the text of Grätzer [6, Chapter IV, §2]. Lemma 2.4. If K is a KR model structure, then the elements of L ( K ) are exactly the non-empty linear subspaces of K . Proof. The lemma follows from the definition of A ◦ B and the fact that Raa 0 and Raaa hold in any KR model structure. Theorem 2.5. If A is a KR -algebra, then L ( K ) , ordered by containment, forms a modular lattice, with least element t , and the lattice operations of join and meet defined by a ∧ b and a ◦ b . Proof. The fact that L ( K ) forms a lattice, with ∧ as the lattice meet and ◦ as the lattice join, can be proved from the basic properties of De Morgan lattices. We now prove modularity; in the following computation, we use juxtaposition ab for meet a ∧ b , and a for the Boolean complement. Note that a ∨ b is the extensional (Boolean) join, not the lattice join in L ( K ). If a ≥ c , then a ( b ◦ c ) = a [( ba ∨ ba ) ◦ c ] a [( ba ◦ c ) ∨ ( ba ◦ c )] = ≤ a [( ba ◦ c ) ∨ ( a ◦ a )] a ( ba ◦ c ) ∨ aa = ≤ ab ◦ c. The opposite inequality ab ◦ c ≤ a ( b ◦ c ) follows from the lattice properties of L ( K ), so a ( b ◦ c ) = ab ◦ c . In the fourth line above, the equation a ◦ a = a follows from Lemma 2.3, since for a ∈ L ( K ), a = a → a , so a = a → a = a ◦ a . The preceding theorem shows that there is a modular lattice canonically asso- ciated with any KR -algebra. It is natural to ask the question: how general is this 4
Geometry of Relevant Implication b v c a c b a ^ c Figure 1: N 5 : the five-element non-modular lattice construction? That is to say, which modular lattices arise in this way? In earlier pa- pers [13],[14],[15] I provided a partial answer to this question by showing that a very large family of modular lattices, closely connected with classical projective geome- tries, can be represented as the lattices L ( K ) associated with KR model structures. This construction made possible the solution of some long-standing problems in the area of relevance logic, particularly those of decidability and interpolability. The lattices arising from projective spaces, however, are of a rather special type, and the construction given in my earlier work does not make clear whether more general modular lattices can be represented. In this section, I give a very simple construction for KR model structures showing that any modular lattice can be represented as a sublattice of a lattice L ( K ). The earlier representation of geometric lattices can be obtained as a direct corollary of this construction, as is shown in Section 4. Definition 2.6. Let L be a lattice with least element 0. Define a ternary relation R on the elements of L by: Rabc ⇔ a ∨ b = b ∨ c = a ∨ c, and let K ( L ) be � L, R, 0 � . Theorem 2.7. K ( L ) is a KR model structure if and only if L is modular. Proof. The first three postulates for a KR model structure follow immediately from the definition of R , using only the fact that L is a lattice. Now assume that L is modular; we need to verify the last postulate (the Pasch postulate). Assume 5
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